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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Generic stationary measures and actions

Authors: Lewis Bowen, Yair Hartman and Omer Tamuz
Journal: Trans. Amer. Math. Soc. 369 (2017), 4889-4929
MSC (2010): Primary 37A35
Published electronically: January 9, 2017
MathSciNet review: 3632554
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Abstract: Let $ G$ be a countably infinite group, and let $ \mu $ be a generating probability measure on $ G$. We study the space of $ \mu $-stationary Borel probability measures on a topological $ G$ space, and in particular on $ Z^G$, where $ Z$ is any perfect Polish space. We also study the space of $ \mu $-stationary, measurable $ G$-actions on a standard, nonatomic probability space.

Equip the space of stationary measures with the weak* topology. When $ \mu $ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of $ (G,\mu )$. When $ Z$ is compact, this implies that the simplex of $ \mu $-stationary measures on $ Z^G$ is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on $ \{0,1\}^G$.

We furthermore show that if the action of $ G$ on its Poisson boundary is essentially free, then a generic measure is isomorphic to the Poisson boundary.

Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when $ G$ has property (T), the ergodic actions are meager. We also construct a group $ G$ without property (T) such that the ergodic actions are not dense, for some $ \mu $.

Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual.

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Additional Information

Lewis Bowen
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712

Yair Hartman
Affiliation: Weizmann Institute of Science, Rehovot, Israel
Address at time of publication: Department of Mathematics, Northwestern University, Evanston, Illinois 60201

Omer Tamuz
Affiliation: California Institute of Technology, Pasadena, California 91125

Keywords: Stationary action, Poisson boundary
Received by editor(s): February 17, 2015
Received by editor(s) in revised form: July 2, 2015, August 10, 2015, and August 14, 2015
Published electronically: January 9, 2017
Additional Notes: The first author was supported in part by NSF grant DMS-0968762, NSF CAREER Award DMS-0954606 and BSF grant 2008274.
The second author was supported by the European Research Council, grant 239885.
Article copyright: © Copyright 2017 American Mathematical Society